While I generally agree, I don’t think of this as out-of-character.
The first part of the journey was hard and dreary, and Frodo remembered little of it, save the wind. For many sunless days an icy blast came from the Mountains in the east, and no garment seemed able to keep out its searching fingers. Though the Company was well clad, they seldom felt warm, either moving or at rest. They slept uneasily during the middle of the day . . .
That’s a brief summary of actions that take many days. They travelled X miles south. It could include “they spoke in Sindarin, so that the hobbits could learn it” or “Frodo’s wound healed over the course of three weeks.” But all of it is about the protagonists, so it seems to me to be still in character. Of course the players are having to figure out game mechanics, but they would have to do that in a fight to work out their chances of hitting their foes, and I don’t think we would call that “out of character.” I think of it, rather, as speaking about the characters and their actions and experiences in a summary mode.
Agree with your analysis, but it seems I have a different definition of “in character”. I’m assuming there is PC dialogue and PC POV for it to be in character. So this is in-character:
Frodo’s player: “I change the dressing on my wound and take vitamin D tablets to promote healing.”
Another player: “Poor Frodo, it must be boring stuck in bed all day. I’ll teach you to speak Sindarin.”
This is out of character:
Frodo’s player: Three weeks? Okay that means 9 hit points back. Do I add the vitamin D bonus? And can I spend my xp to learn Sindarin?
In reality, players probably dip in and out of both those modes in a downtime chunk.
I think I prefer a different use of “in character.” For me, “out of character” would mean what I’ve seen called a “cut scene”; for example, at the very end of my Manse campaign, the guest player who played the adversary (the spirit of a long dead necromancer, who had been haunting the caverns under the castle, and whose remains a group of player characters had sought out and performed rites of passage over) had a final scene where he said to one of his servants that it was a good thing he had thought to prepare a fake body for himself. That was something none of the player characters was aware of. In contrast, both of your portrayals are things that Frodo and the other character would be aware of. The presentation is different—sort of like the difference between third person and omniscient PoV—but the experience being narrated is not. (When the narrator of Monty Python and the Holy Grail says “They were forced to eat Sir Robin’s minstrels, and there was much rejoicing,” that doesn’t take place on camera for obvious reasons, but it’s still part of the experience of King Arthur and his knights.)
I agree that both distinctions are meaningful; it’s just a difference of preferred terminology.